Seismic waves generated by an impulsive point source in a solid/fluid configuration with a plane boundary

Geophysics ◽  
1985 ◽  
Vol 50 (7) ◽  
pp. 1083-1090 ◽  
Author(s):  
Adrianus T. de Hoop ◽  
Jos H. M. T. van der Hijden
Keyword(s):  
Point Source ◽  
Seismic Waves ◽  
Integral Transform ◽  
Vertical Plane ◽  
Computation Time ◽  
Plane Boundary ◽  
Head Wave ◽  
Transform Method ◽  
Expansion Model ◽  
Fluid Configuration

The space‐time acoustic wave motion generated by an impulsive point source in a solid/fluid configuration with a vertical plane boundary is calculated with the aid of the modified Cagniard method. Two types of sources are considered in detail, viz. (1) a point source of expansion (model for an explosive source), and (2) a point force parallel to the vertical interface (model for a mechanical vibrator). Numerical results are presented for the transmitted scalar traction in the fluid in those regions of space where head wave contributions occur. There is a marked difference in the time response observed for the two types of sources and for the different positions of the receiver in the fluid with respect to the position of the source in the solid. These waveform differences are important when the transmitted wave in the fluid is used to determine experimentally the elastic properties of the solid. Scholte waves are observed only when the source is close to the fluid/solid interface. As compared with the traditional Fourier‐Bessel integral transform method of handling this problem, the computation time with the method presented here is considerably less.

Numerical evaluation of the transient acoustic waveform due to a point source in a fluid‐filled borehole

Geophysics ◽  
1979 ◽  
Vol 44 (10) ◽  
pp. 1706-1720 ◽  
Author(s):  
Leung Tsang ◽  
Dennis Rader
Keyword(s):  
Numerical Methods ◽  
Point Source ◽  
Numerical Evaluation ◽  
Close Agreement ◽  
Time Windows ◽  
Computation Time ◽  
Head Wave ◽  
Oil Well ◽  
Additional Information ◽  
Frequency Domains

A key measurement employed in oil well wireline logging is the acoustic wave traveltime over a specified formation interval, typically 1 ft. In the traditional measurement, only the compressional head wave is monitored, but for some time it has been obvious that there is significant additional information, such as the shear head wave arrival, in the received waveform. We describe two numerical methods for computing the profile and parameter dependence of the transient waveform based on a model of the acoustic logging problem consisting of a point source on the axis of a fluid‐filled cylindrical borehole. The response to this excitation is determined at a distance from the source, generally on the borehole axis. In the first of the two numerical methods, called “real axis integration”, the complete acoustic waveform is obtained. The second method, called “branch‐cut integration”, evaluates the first compressional and shear‐pseudo‐Rayleigh arrivals individually with much less computation time than the first method. The validity and accuracy of the two methods are demonstrated by their close agreement within appropriate time windows. It is also shown that the results from the ordinary asymptotic method that exist in the literature predict different behavior. The dependence of the transient arrivals on formation parameters is illustrated by various numerical results in both time and frequency domains.

scholarly journals Semi-analytical solutions of seismo-electromagnetic signals arising from the motional induction in 3-D multi-layered media: part II—numerical investigations

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Hengxin Ren ◽  
Ling Zeng ◽  
Yao-Chong Sun ◽  
Ken’ichi Yamazaki ◽  
Qinghua Huang ◽  
...  
Keyword(s):  
Point Source ◽  
Numerical Computation ◽  
Wave Velocity ◽  
Excellent Agreement ◽  
Seismic Waves ◽  
Averaging Method ◽  
Ground Surface ◽  
Induction Effect ◽  
Motional Induction ◽  
Electromagnetic Signals

AbstractIn this paper, numerical computations are carried out to investigate the seismo-electromagnetic signals arising from the motional induction effect due to an earthquake source embedded in 3-D multi-layered media. First, our numerical computation approach that combines discrete wavenumber method, peak-trough averaging method, and point source stacking method is introduced in detail. The peak-trough averaging method helps overcome the slow convergence problem, which occurs when the source–receiver depth difference is small, allowing us to consider any focus depth. The point source stacking method is used to deal with a finite fault. Later, an excellent agreement between our method and the curvilinear grid finite-difference method for the seismic wave solutions is found, which to a certain degree verifies the validity of our method. Thereafter, numerical computation results of an air–solid two-layer model show that both a receiver below and another one above the ground surface will record electromagnetic (EM) signals showing up at the same time as seismic waves, that is, the so-called coseismic EM signals. These results suggest that the in-air coseismic magnetic signals reported previously, which were recorded by induction coils hung on trees, can be explained by the motional induction effect or maybe other seismo-electromagnetic coupling mechanisms. Further investigations of wave-field snapshots and theoretical analysis suggest that the seismic-to-EM conversion caused by the motional induction effect will give birth to evanescent EM waves when seismic waves arrive at an interface with an incident angle greater than the critical angle θc = arcsin(Vsei/Vem), where Vsei and Vem are seismic wave velocity and EM wave velocity, respectively. The computed EM signals in air are found to have an excellent agreement with the theoretically predicted amplitude decay characteristic for a single frequency and single wavenumber. The evanescent EM waves originating from a subsurface interface of conductivity contrast will contribute to the coseismic EM signals. Thus, the conductivity at depth will affect the coseismic EM signals recorded nearby the ground surface. Finally, a fault rupture spreading to the ground surface, an unexamined case in previous numerical computations of seismo-electromagnetic signals, is considered. The computation results once again indicate the motional induction effect can contribute to the coseismic EM signals.

Dynamic properties of reflected and head waves near the critical point

1979 ◽  
Vol 16 (7) ◽  
pp. 1388-1401 ◽  
Author(s):  
Larry W. Marks ◽  
F. Hron
Keyword(s):  
Exact Solution ◽  
High Frequency ◽  
Classical Problem ◽  
Plane Boundary ◽  
Head Wave ◽  
Disk File ◽  
Ray Theory ◽  
Computational Point ◽  
Spherical Waves ◽  
Head Waves

The classical problem of the incidence of spherical waves on a plane boundary has been reformulated from the computational point of view by providing a high frequency approximation to the exact solution applicable to any seismic body wave, regardless of the number of conversions or reflections from the bottoming interface. In our final expressions the ray amplitude of the interference reflected-head wave is cast in terms of a Weber function, the numerical values of which can be conveniently stored on a computer disk file and retrieved via direct access during an actual run. Our formulation also accounts for the increase of energy carried by multiple head waves arising during multiple reflections of the reflected wave from the bottoming interface. In this form our high frequency expression for the ray amplitude of the interference reflected-head wave can represent a complementary technique to asymptotic ray theory in the vicinity of critical regions where the latter cannot be used. Since numerical tests indicate that our method produces results very close to those obtained by the numerical integration of the exact solution, its combination with asymptotic ray theory yields a powerful technique for the speedy computation of synthetic seismograms for plane homogeneous layers.

scholarly journals On the point-source approximation of earthquake dynamics

2014 ◽  
Vol 57 (3) ◽  
Author(s):  
Andrea Bizzarri
Keyword(s):  
Point Source ◽  
Seismic Waves ◽  
Single Point ◽  
Seismic Source ◽  
Transition Process ◽  
Point Sources ◽  
Multiple Point ◽  
Earthquake Dynamics ◽  
Planar Fault ◽  
Source Models

<p>The focus on the present study is on the point-source approximation of a seismic source. First, we compare the synthetic motions on the free surface resulting from different analytical evolutions of the seismic source (the Gabor signal (G), the Bouchon ramp (B), the Cotton and Campillo ramp (CC), the Yoffe function (Y) and the Liu and Archuleta function (LA)). Our numerical experiments indicate that the CC and the Y functions produce synthetics with larger oscillations and correspondingly they have a higher frequency content. Moreover, the CC and the Y functions tend to produce higher peaks in the ground velocity (roughly of a factor of two). We have also found that the falloff at high frequencies is quite different: it roughly follows ω<span><sup>−2</sup></span> in the case of G and LA functions, it decays more faster than ω<span><sup>−2</sup></span> for the B function, while it is slow than ω<span><sup>−1</sup></span> for both the CC and the Y solutions. Then we perform a comparison of seismic waves resulting from 3-D extended ruptures (both supershear and subshear) obeying to different governing laws against those from a single point-source having the same features. It is shown that the point-source models tend to overestimate the ground motions and that they completely miss the Mach fronts emerging from the supershear transition process. When we compare the extended fault solutions against a multiple point-sources model the agreement becomes more significant, although relevant discrepancies still persist. Our results confirm that, and more importantly quantify how, the point-source approximation is unable to adequately describe the radiation emitted during a real world earthquake, even in the most idealized case of planar fault with homogeneous properties and embedded in a homogeneous, perfectly elastic medium.</p>

Reflection of an Acoustic Step Wave From an Elastic Cylinder

1958 ◽  
Vol 25 (1) ◽  
pp. 103-108
Author(s):  
Richard Skalak ◽  
M. B. Friedman
Keyword(s):  
Integral Transform ◽  
Formal Solution ◽  
Elastic Cylinder ◽  
Transform Method ◽  
Pressure Fields ◽  
Plane Step ◽  
Single Integral ◽  
Short Time ◽  
Acoustic Fluid

Abstract An elastic cylinder, circular in section and infinite in length, is considered in an infinite acoustic fluid. The object of this paper is the determination of the reflected and diffracted pressure fields at large distances resulting from a plane step wave of pressure impinging on the cylinder and moving in a direction normal to the axis of the cylinder. A formal solution is obtained for the general case of an elastic cylinder. Numerical results are computed for rigid, fixed cylinders, and for rigid, floating cylinders. Two different methods are used to achieve results in the different ranges of time which are of interest. A short time approximation is developed by the use of a double integral-transform method. A mode approach and a single integral transform are used for later times. The results show that the reflected pulse decays quickly, within a time on the order of the transit time of the original wave across the cylinder.

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